Furstenberg correspondence principle

Statement

For $E\subseteq\mathbb Z$ , there exist an invertible probability-preserving system $(X,\mu,T)$ and a measurable event $A$ with $\mu(A)=d^*(E)$ such that, for every finite list $n_1,\ldots,n_k\in\mathbb Z$ ,

\mu(A\cap T^{-n_1}A\cap\cdots\cap T^{-n_k}A) \leq d^*\bigl(E\cap(E-n_1)\cap\cdots\cap(E-n_k)\bigr).

Here $E-n=\{m:m+n\in E\}$ .

Construction

Encode $E$ by its binary indicator sequence in $\{0,1\}^{\mathbb Z}$ . On the closure of its shift orbit, average point masses over intervals where the density approaches $d^*(E)$ , and take a weak limit. Boundary terms vanish relative to interval length, so the limit is shift invariant. The coordinate event $x_0=1$ gives $A$ ; finite-coordinate intersections give the inequality.

Scope

This version preserves the density exactly but does not promise an ergodic model. It is sufficient for arithmetic recurrence because the multiple recurrence theorem does not require ergodicity.

References

Asgar Jamneshan and Henrik Kreidler, Ergodic Structure Theory and Applications, ISEM 28 lecture notes. Theorem 4.2.9, specialized to interval averages in the integers.